The distance to the nearest stars can be calculated by observing how their positions shift, when viewed six months apart, against the background of more distant stars. This is parallax. Here is the apparent position of a nearby star viewed against the background of more distant start. Here is the apparent position six months later. Here is the shift in apparent position. Proxima Centauri, the nearest star (other than the sun), is 4.25 light years from earth and has an observed shift in position of 0.76 seconds of arc, much too small to be measured without a telescope. The first determination of distance by measuring paralax was made by F. W. Bessel in 1837. For the star 61 Cygni, Bessell measured a parallax of 0.293 seconds of arc.
In this way, we can measure the distances to about 100 light years. How can we determine the dsitance to stars farther away?
If we knew how bright a star is, we could determine how far away it is. The precise relation between distance, apparent brightness, and true brightness is an elaboration of a common-sense observation: given two lights of the same brightness, the one farther away will appear dimmer.
For the nearby stars, distance can be determined by parallax. Knowing how bright they appear and how far away they are, we can determine the actual brightness of these nearest stars. For these stars, Hertzsprung and Russell compared the actual brightness to the color (more precisely, the spectral type) of the stars. The stars fit into an approximately diagonal band called the main sequence. This graph is called the HR Diagram. With it, the color of a star can be related to its actual brightness. Comparison with its apparent brightness allows the distance to be determined. This works reasonably well to determine the distance to stars, but what about the distance to galaxies? In other galaxies, most individual stars can't be resolved well enough to determine spectral type, so the HR disgram can't be used.
While the luminosity of the sun is relatively constant, some stars undergo substantial oscillations in brightness. These are called variable stars. One type, Cephid variables, exhibit a relationship between the period of their oscillations and their absolute brightness. At their brightest, Cephids can be seen in nearby galaxies. So the distance scale to the galaxies was set.
In 1929 E. Hubble discovered that the light of distant galaxies is redshifted. The Doppler shift toward longer wavelengths indicates the source is moving away from us. The familiar acoustical version is the drop in pitch of a siren as the ambulance rushes past us. Hubble deduced the universe is expanding. (This is a consequence of Einstein's original cosmological model, but Einstein rejected it as philosophically unacceptable.) Indeed, Hubble's law is that the velocity V of recession of galaxies is proportional to their distance R from us.
V = HR
The constant of proportionality is called Hubble's constant; measurements of Hubble's constant are the basis of estimates of the age of the universe.
All these measurements involve uncertainties, perhaps the largest source being interstellar and intergalactic dust. Because each type of measurement sets the baseline for the next, errors compound, reducing the certainty of our estimates of large distances.
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