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1.3.4 Measurement of Distance

In observing stars, the observer knows only the direction from which the star's photons are arriving at his telescope on earth. This direction may change as the earth moves in its orbit around the sun, which provides a means of measuring distance by triangulation. However, most stars are so far away that the triangulation angles are too small to be measured accurately. One way to improve measurement of these angles is to measure them relative to the very distant 'fixed' stars. This is known as parallax triangulation.

In parallax triangulation, two parallax angles are measured six months apart, so that photon trajectories of these two measurements are on two different plane sections. The
triangulation angle obtained by this method is the angle at which the two planes intersect along the radius vector of the star. This leads to an over estimation of distance, $r_o$, to the distant star. The geometry of the situation indicates that parallax distance is given by \(r_o = R_o\ tan \theta\), whereas, the true distance, \(r_o = R_o \theta\). However, even parallax measurements of distance are only possible for small distances in which \(\theta\) and \(tan \theta\) are substantially equal. Hence, parallax measurements are valid within the accuracy of measurement.

For stars and galaxies which are very far away, distance can be estimated by their relative brightness assuming that it is diminished according to an inverse square law. For this to succeed it is necessary to identify stars which have a known luminosity, such stars are called 'standard candles'. However, this method is also in error since the curvature of the continuum modifies the inverse square law. Consider a star emitting photons in all directions. As the photons travel through the energy continuum, at a radius, \(r_p\), the surface area of the radiating sphere will not be \(4\pi r^2\) because of the curvature of the continuum. The brightness of the star will be greater than in a flat universe. This leads to an underestimation of the star's distance by a factor \((sin\ \theta/\theta)^2\), where \(\theta\) is the angle of separation of the star from the Solar system.

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