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As light travels away from its source star S, it spreads out and weakens in intensity. Moving out from a distance of r to a distance of 2r, the light that previously covered a square of area r2 spreads out to cover an area of 2r × 2r = 4r2. When the same light reaches a distance of 3r, it covers an area of 3r × 3r = 9r2. The intensity is inversely proportional to the area covered, which expands with the square of the distance, a rule known as the inverse square law.


The inverse square law

Newton tried to estimate the distances of the stars using the inverse-square law for light. Suppose that a star has luminosity L watts, so the total power of the light radiated from the surface of the star in all directions is L. At distance r (metres) from the star, this light is spread out over a sphere of area 4πr2 (m2).

So at a distance r from the star the intensity of radiation I that we measure in a detector, for example the human eye, is I = L / 4πr2 (watts m–2). Since I is proportional to the inverse square of the distance, this is called the inverse-square law for light (see figure above).

Newton did not have a satisfactory way of estimating the intensities of light from stars, so he did not get sensible results, but this problem was solved during the 19th century. The ancient Greeks classified the stars visible to the naked eye on a magnitude scale from 1 to 6, where the brightest stars are magnitude 1 and the faintest magnitude 6. This system turned out to be an approximately logarithmic scale, with a change of about 2.5 for each change of 1 magnitude, so the stellar magnitude scale was defined as

m = constant – 2.5 log10 I

where the constant depends on the observing waveband. (The apparent brightness of a star will appear different depending on the filter – red, yellow, blue, etc – used to observe it.) 


Parallax is the small change in direction of nearby stars with respect to much more distant stars that results from the motion of the Earth around the Sun. Parallax allows us to estimate the distances of thousands of nearby stars. The parallax angle p is half the apparent change in position of the star at a distance d, with respect to more distant stars, seen in separate observations six months apart.


The first step on the distance ladder beyond the solar system was taken by the German astronomer Friedrich Wilhelm Bessel in 1838. He measured the parallax of the nearby star 61 Cygni – that is, the change in its apparent direction on the sky resulting from the movement of the Earth in its orbit around the Sun. This parallax phenomenon was the final proof of the Copernican system. Astronomers adopted a unit of distance related to parallax, the parsec. One parsec (1 pc) is the distance of a star when the half-angle subtended by the Earth’s orbit at the star is one second of arc.

In 1781 the French comet-finder Charles Messier made a list of 110 objects which looked fuzzy or extended through a small telescope. Using much larger telescopes the British astronomer (and discoverer of Uranus) William Herschel showed that many of these are in fact clusters of stars.

In the 19th century astronomers used prisms and then gratings to disperse the light from astronomical objects into a spectrum. They found that many spectra had characteristic dark (absorption) or bright (emission) lines superimposed on them which indicated the presence of particular elements. This spectroscopy technique showed that some of the fuzzy objects were hot clouds of gas heated by young stars.

The nature of one remaining class of nebulae, the spiral nebulae, remained uncertain. Were they simply gas clouds in our Milky Way system, or could they be distant island universes (vast collections of stars now known as galaxies), as suggested by Christopher Wren, Thomas Wright and Immanuel Kant?


Doppler shift

The advent of spectroscopy also allowed measurement of the velocity of stars in the line of sight, via the Doppler shifting of emission and absorption lines in their spectrum.

The Austrian physicist Christian Doppler made his discovery using sound waves in 1842. If a source emitting sound of wavelength λ is moving away from us, the wavelength λ will be increased by ∆λ = v/s, where v is the velocity of the source away from us and s is the speed of sound. Similarly a source moving towards us has its wavelength shortened by the same amount. This happens because the waves get bunched together or stretched apart by the motion of the source relative to the observer.

For light the shift is ∆λ = v/c, where c is the speed of light and the redshift or blue shift z = ∆λ/λ.

The Doppler studies of stars in our galaxy showed that in addition to the broad circular motion of stars expected in a rotating disc, there were also random motions up and down, which broaden out the disc. So our galaxy is different from the solar system, where the planetary orbits are confined near a single plane.

Looking at the spectra of galaxies, astronomers can use this technique to measure the velocity at which they are receding from or sometimes approaching the Earth.

The Doppler shift, in which the wavelength of light (or sound) is shifted, depending on whether a source is approaching or receding. If a source is approaching the observer, the characteristic emission or absorption lines are at a shorter wavelength and shifted towards the blue end of the spectrum (blue shift). If the source is moving away, the lines are shifted to a longer wavelength towards the red end of the spectrum (redshift).


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