On the coloured light of the binary stars and some other stars of the heavens

Albireo, a well-known coloured double star. Compare the colour of other stars in [1]

On the coloured light of the binary stars and some other stars of the heavens or in the original German Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels is a treatise by Christian Doppler (1842)^[1] in which he postulated his principle that the observed frequency changes if either the source or the observer is moving, which later has been coined the Doppler effect. The original German text can be found in wikisource. The following annotated summary serves as a companion to that original.

Title

The title "Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels - Versuch einer das Bradley'sche Aberrations-Theorem als integrirenden Theil in sich schliessenden allgemeineren Theorie" (On the coloured light of the binary stars and some other stars of the heavens - Attempt at a general theory including Bradley's theorem as an integral part) specifies the purpose: describe the hypothesis of the Doppler effect, use it to explain the colours of binary stars, and establish a relation with Bradley's stellar aberration.^[2]

Content

§ 1 In which Doppler reminds the readers that light is a wave, and that there is debate as to whether it is a transverse wave, with aether particles oscillating perpendicular to the propagation direction. Proponents claim this is necessary to explain polarised light, whereas opponents object to implications for the aether. Doppler doesn't choose sides, although the issue returns in § 6.

§ 2 Doppler observes that colour is a manifestation of the frequency of the light wave, in the eye of the beholder. He describes his principle that a frequency shift occurs when the source or the observer moves. A ship meets waves at a faster rate when sailing against the waves than when sailing along with them. The same goes for sound and light.

§ 3 Doppler derives his equations for the frequency shift, in two cases:

		Equation Doppler^[3]	Modern equation
1.	Observer approaching stationary source with speed v_o	$n/x=(a+\alpha _{\rm {o}})/a$	$f'/f=(c+v_{\rm {o}})/c$
2.	Source approaching stationary observer with speed v_s	$n/x=a/(a-\alpha _{\rm {s}})$	$f'/f=c/(c-v_{\rm {s}})$

§ 4 Doppler provides imaginary examples of large and small frequency shifts for sound:

v_o = -c	f' = 0	frequency shift down to inaudibly low tones
v_s = -c	f ' / f = 0.5	frequency shift down over 1 octave, still audible.
v_s = +c	f ' / f = ∞	frequency shift up to inaudibly high tones^[4]
v_o = 40 m/s^[5]	C to D	note C shifting to D.
v_o = 5.4 m/s	quarter note	threshold for best observers with absolute hearing^[6]

§ 5 Doppler provides imaginary examples of large and small frequency shifts for light from stars. Velocities are expressed in Meilen/s, and the light speed has a rounded value of 42000 Meilen/s.^[7] Doppler assumes that 458 THz (extreme red) and 727 THz (extreme violet) are the borders of the visible spectrum,^[8] that the spectrum emitted by stars lies exactly between these borders (except for the infrared stars of § 8), and that the colour of the light emitted by stars is white.^[9]

Meilen/s	km/s	f ' / f
v_s = -19000^[8]	141000	458 / 727	shift from extreme violet to extreme red, and from other colours to invisible range beyond extreme red^[10]
v_s = -5007^[8]	37200	458 / ?	shift from yellow to extreme red
v_s = -1700	12600	458 / ?	shift from red to extreme red
v_s = -33	244	458 / 458.37	threshold for visual perception of colour changes^[11] shift from a shade of red to next shade of red approaching white star gets a green shade receding white star gets an orange shade

§ 6 Doppler summarises:

The natural colour of stars is white or a weak yellow.
A white star approaching with progressive speed would successively turn to green, blue, violet, and invisible (ultraviolet).
A white star receding with progressive speed would turn to yellow, orange, red, and invisible (infrared).

Doppler wishes that his frequency shift theory will soon be tested by another method to determine the radial velocity of stars. He thinks, without reason, that a confirmation of his theory would imply that light is not a transverse but a longitudinal wave.^[12]

§ 7 Doppler argues that his theory applies mainly to binary stars. In his opinion the fixed stars^[13] are immobile and white.^[14] In a binary star high speeds could be possible due to orbital motion,^[15]^[16] and binaries appear to be colourful.^[17] Doppler divides the binaries in two groups: (1) binary stars of unequal brightness; and (2) binary stars of equal brightness. His interpretation is: in case (1) the brighter star is the heavier one, the weaker star revolves around him; in case (2) both stars revolve around a center of mass in the middle, or around a dark third star. In case (2) the colours are usually complementary. Doppler rules out that the rich complementary colours of binaries are contrast illusions, because an astronomer said he had observed that covering one star does not change the colour impression of the other star. Doppler claims that his theory is supported by the fact that for many binary stars the colour indication in Struve's catalogue is different from that in Herschel's older catalogue, attributing the difference to progress of the orbital motion.^[18]

§ 8 Doppler presents two groups of variable stars that in his opinion can be explained as binary stars with Doppler effect. These are the "other stars in the heavens" from the title.

Periodic variable stars that are invisible for most of the time, and that brighten up with a red colour for a short while once per cycle. In Doppler's opinion they are binary stars. Such a star is usually invisible because it emits infrared instead of white light. In the orbit section with the maximal radial speed in the direction of Earth, the observed frequency on Earth is shifted from infrared to visible red.
'New stars' (in particular two supernovas, Tycho's Nova of 1572, and Kepler's Nova of 1604), that suddenly appeared, having a white colour in the brightest phase, then turning to yellow and red, and finally fading out. According to Doppler they too are binary stars, with extremely high speed and long period.^[19] Doppler assumes Sirius, the brightest star in the sky, belongs to this group, because some texts from antiquity say its colour was red, instead of its current white colour.^[20]

§ 9 Doppler notes that the orbital speed of the Earth (4.7 Meilen/s) is too low (<33 Meilen/s) to result in visually perceptible colour changes. He identifies two factors that may lead to high orbital speeds in a binary star:

Central star far heavier than the Sun. According to Doppler stars that are a million times heavier than the Sun are plausible.^[21]
Highly elliptical orbit with a small perihelium distance^[22] (<1 AU).

Doppler assumes that there are binary stars with a perihelium speed larger than the speed of light. The astronomer Littrow would have suggested that the perihelium speed of the visual binary star γ Virgo is nearly equal to the speed of light.

§ 10 Doppler summarises the above, and concludes that his speculations explain so much that his theory has to be true. He shares a few more speculations:

The colours of binary stars are not static, they will change periodically in phase with the orbital motion.
The stars of § 8, which suddenly (in just a few hours time) appear, then gradually extinguish and remain invisible for many years, are binary stars with a highly elliptical orbit and a high perihelium speed. If the Earth sees the orbit obliquely, such a star may appear faster than it disappears.
Fluctuations in the period of variable stars like Mira (according to Doppler its period varies between 328 and 335 days), result from the orbital motion of the Earth.

§ 11 Conclusion: Doppler expects his frequency shift theory will be accepted, because similar aberrations that depend on v/c (Rømer's and Bradley's)^[23] have been accepted before. Doppler waits for the experts to decide if his speculations will do as evidence. He is convinced that finally his principle will be used for the determination of the speed of remote stars.^[24]

Notes

^ Some sources mention 1843 as year of publication because in that year the article was published in the Proceedings of the Bohemian Society of Sciences. Doppler himself referred to the publication as "Prag 1842 bei Borrosch und André", because in 1842 he had a preliminary edition printed that he distributed independently.
^ In 1728 Bradley discovered and explained the so called aberration of star light. This aberration was one of the first compelling pieces of evidence for the finite speed of light in the universe. Finite meaning in this case: although large, not extremely large compared to the orbital speed of the Earth. Bradley's aberration is approximately proportional to v/c, the ratio of the speed of the Earth over the speed of light. The Doppler effect contains a similar proportionality to v/c.
^ Doppler uses other symbols for the variables than we usually do today: f = 1/n, f ' = 1/x, v_o = α_o, v_s = α_s. (Note n=n" and x=x", as in number of seconds = time).
^ inaudible, except for the shock wave ignored by Doppler.
^ 1 par.Fuss = 0.325 m (pied de roi); speed of sound 1024 par.Fuss/s = 333 m/s
^ In 1845 Buys Ballot used this idea of musicians with absolute hearing for the first experimental verification of the Doppler effect.
^ Meile = geografische Meile = 7420 m. Doppler provides the rounded value of 42000 Meilen/s instead of the best accurate value of his time. The rounded value was well known and stable throughout the years, whereas the accurate value varied due to frequent new measurements. Since 1835 the accurate value was 41549 geogr. Meilen/s (308000 km/s), see Pierer's Universallexikon and Wüllner's Experimentalphysik

^ ^a ^b ^c The frequencies 458 THz (extreme red) and 727 THz (extreme violet) and other colours apparently have been derived from the wavelengths mentioned by Thomas Young in his Theory of Light and Colours (1802), where Doppler would have used 309000 km/s as light speed. This explains most values in the table, except for the v_s-values 19000 and 5007 that remain as calculation errors by Doppler (deviation about 25%).

Colour	Wavelength according to Young (nm)	Frequency if c=309000km/s (THz)	v_s=c(1-f/f') if c=309000km/s (geogr. Meilen/s)	v_s according to Doppler (geogr. Meilen/s)
extreme violet	425	727	-24462	-19000 (error)
yellow	577	535	-7037	-5007 (error)
red	648	477	-1704	-1700
extreme red	675	458	0	0

^ These assumptions are wrong. Doppler ignores the emitted infrared and ultraviolet, although their presence in sun light was known since studies by Herschel (1800) and Ritter (1801). As a result Doppler overstimates the visual colour changes. He knew that stars are able to emit infrared, as he proposes so in § 8. With regard to the colours of stars, the assumption that stars emit white light is his major mistake. Nowadays we know that colour mainly depends on the star temperature.
^ Doppler uses the old term homogeneous light for monochromatic light.
^ Derivation of the threshold value 458.37: Herschel stated that white light, obtained by mixing red, yellow and blue light, undergoes a visually perceptible colour change if the intensity of any of these three components changes by at least 1%. According to Doppler this implies that white star light undergoes a perceptible colour change if the frequency shift is at least 1% of the red segment of the spectrum. Using Young's definition of the red segment (wavelength 625 - 675nm, Theory of Light and Colours, see above) and c=309000km/s (see above), the red segment corresponds to the frequency range 458 - 495THz. The value in that segment at 1% distance from the border is 458.37THz.
^ Bolzano, in his review in 1843, points out that Doppler's thought that his theory would not apply for transverse waves is a mistake. Annalen der Physik 1843
^ In Doppler's article, 'fixed stars' are single stars that are not part of a binary star. The idea of their immobility is a legacy from antiquity, when the ideal fixed stars were contrasted with the planets.
^ Bolzano argues in his review in 1843 that the idea of the immobility of single stars is unnecessary, and that the observed proper motion of many stars indicates that single stars do move. Annalen der Physik 1843
^ Later it turned out that the orbital speed of binary stars is not particularly large compared to the proper motion speed of stars. Speeds up to about 200 km/s are observed in eclipsing binaries. An exception is the fastest binary star, a rare type of two white dwarfs, with a period of 5 minutes, an orbit diameter of 80000 km, and an orbital speed of more than 1000 km/s. RX J0806.3+1527
^ At present, the highest radial speed of nearby single stars is about 300 km/s. (Doppler effect#Astronomy)
^ With a few known facts Doppler could have easily estimated that the orbital speed of the visual binaries, which appear to be colourful, is smaller than the orbital speed of the Earth. The distance from the Earth to stars is at least 4 light years (that is the distance to the nearest star). A traditional telescope at sea level has a resolving power of 1 arc second or worse, due to atmospheric turbulence. Hence, the two stars that compose a visual binary have a distance of at least 1 AU. The shortest period of a visual binary is 1.7 year. Therefore the orbital speed of visual binaries (with a circular orbit) is smaller than that of the Earth, below the threshold for visual colour changes (see § 9). That is another flaw in Doppler's explanation of the colours of visual binaries.
^ An additional motive for Doppler to focus on binary stars might have been that binary stars were a hot topic in astronomy. Accurate binary star catalogues had been composed by Herschel and by Struve. It had been discovered that binary stars are not static, but that they revolve in an orbit around a center, bound by gravity. The orbit parameters (speed, period, and excentricity) were determined. It had become clear that visually single, variable stars with a particular brightness development, were actually binary stars (eclipsing binaries, like Algol).
^ Therefore Doppler expects that supernovas flame periodically.
^ For details see Sirius Red controversy. This idea meant that Sirius would be a binary star with an extremely long period and high speed. This is incorrect: although Sirius actually is a binary star (as discovered in 1844), it doesn't have a high speed.
^ We now know that the heaviest stars are 100 times heavier than the Sun, but a black hole could be a million times heavier than the Sun. See solar mass.
^ In the case of binary stars the perihelium should actually be called periastron
^ Doppler's suggestion that his theory encompasses Bradley's aberration is an exaggeration. However, he could have claimed that his theory encompassed Ole Rømer's (among astronomers equally famous) aberration of the revolution of Jupiter's moon Io, which Rømer used in 1676 to determine the finite speed of light. That aberration is exactly given by f ' / f = (c+v_o) / c, where f ' and f are the apparent and the actual frequency of revolution. In addition, this shows that the Doppler effect applies to more than just the oscillation frequency of a wave.
^ In his time, Doppler could have thought himself of measuring the shift of spectral lines of stars, although he didn't. In 1815 Fraunhofer had observed dark lines in spectra of the Sun and Sirius. He proposed that every star has a unique line spectrum. A few years later he measured the wavelength of these lines, using a grating. In 1823 William Herschel suggested that the chemical composition of stars could be derived from their spectrum. In 1848 Fizeau pointed to the possibility of measuring the shift of spectral lines in star spectra. But until the breakthrough of the work by Kirchhoff and Bunsen in 1859, spectroscopy remained a difficult method, producing complex and fairly useless spectra. In 1868 Huggins discovered a redshift in the spectrum of Sirius, and he calculated the speed. In 1871 Vogel succeeded in measuring the shift of spectral lines at the edges of the Sun, and he used it to calculate the rotation speed of the sun. In the same year Talbot pointed to the possibility of discovering spectroscopic binary stars by means of periodic doubling of spectral lines, and in 1889 this was observed for the first time in the star Mizar A, by Pickering. See The rise of astrophysics

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