# Cosmic Distance Ladder

By

Dr Lim Zhi Han

,

Koh Zhi Yong

,

Gerald Kang Joon Kiat

●

Published 2022-09-10

The universe is a very large place. We have one star, the Sun, in our solar system. Our solar system is part of our galaxy, the Milky Way. Our galaxy is a part of yet even larger groups of galaxies. To get a sense of the scale, watch 3:49 to 7:30 of the following video.

Suffice to say, galaxies contain many stars, and the universe contains many galaxies. All of these entities are extremely far apart from each other by human standards. As a result, conducting distance measurements is extremely challenging. Here, we look at methods used to measure distances in cosmology.

Recall Kepler's third law which we have learn in the first chapter. It provides us with a simple formula to determine the relative size of the planetary orbits of different planets. By knowing the distance from Earth to our Sun as well as the corresponding periods of the planets by observation, one can in principle deduce the absolute scale of our solar system.

As promising as it may sound, the distance from Earth to our Sun was not known during the time of Kepler. The Astronomical Unit (A.U.) is defined as the average distance between Earth and our Sun. It was first determined during the transit of Venus in 1761 with the help of a method proposed by the English astronomer Edmund Halley (although he never lived to see the remarkable phenomenon himself). Below we shall detail the simplified version of the method.

The general scheme of the method involves the use of parallax and requires two observers at different locations of Earth to record the transit independently before comparing their results.

The figure above shows the schematic of the transit of Venus as observed from two different locations on Earth. Details of how the astronomical unit is derived from trasit-of-Venus observations can be found in the Appendix.

The distance between us and nearby stars can be found by the stellar parallax method as sketched in the figure below. The star of interest is observed twice, with the second observation made six months after the first. The (different) positions of the star with respect to the fixed stars in the background were recorded. The reciprocal of the parallax angle (half the change in angular position) gives the distance of the star from Earth.

d=\frac{\text{1 A.U. }}{\tan p\;(\text{radians})}\approx\frac{\text{1 A.U. }}{p\;(\text{radians})}=\frac{1}{p'"\;(\text{arc-seconds})}\text{pc }

1\text{ radian}=206264.8''\text{ (arcseconds)}

1\text{ pc}=2.062648\times10^{5}\text{ A.U. }=3.08568\times10^{16}\text{ m}

An independent method of distance measurement is required before we can assign absolute magnitudes to the stars we see in the sky. One such method relies on observing a type of star called a Cepheid variable. Cepheid variables are stars that contract and expand in size periodically, brightening and dimming periodically in the process. The first discovery of such an object was o Ceti in 1595. The brightness of this star varied over five orders of magnitudes with irregular periods of between 100 to 200 days. In 1784, a pulsating star with a regular period of 5 days and 9 hours was discovered. Other similar pulsating stars were subsequently found and were called Cepheid variables. Cepheids pulsate because the competing forces of gravity (inwards) and pressure (outwards) are far from equilibrium, similar to a wildly swinging pendulum.

Several discoveries related to the properties of Cepheid variables made it possible to use them as independent measures of distance. The first discovery was a empirical relation between the pulsation period and the perceived brightness of the stars. Henrietta Swan Leavitt (1868 - 1921) found more than 2000 Cepheid variables, most of them located in the dwarf galaxy called the Small Magellanic Cloud (SMC). In 1912, she used a sample of 25 Cepheids in the SMC to show that the apparent magnitude of brightness varies linearly with the logarithm of the period of pulsation.

Since the SMC has a diameter of about 7000 light years and its distance from us is much greater at about 200,000 light years, the stars in the SMC can be considered to be approximately the same distance from us. Hence, the linear relation between absolute magnitudes and the logarithm of their pulsation periods of Cepheids also holds.

The second discovery provided a zero point to calibrate Leavitt's linear relation. This was first carried out in 1913 by Ejnar Hertzsprung (1873 - 1967), who measured the absolute distances of a few nearby Cepheid variables via statistical parallax, a modified stellar parallax technique that used the motion of the sun as a longer baseline together with measurements of average velocities of stars. Comparing the absolute distance with the apparent magnitude of a Cepheid variable, it was thus possible to determine its absolute magnitude. The absolute magnitude could then be correlated with its pulsation period, giving a reference point to use in conjunction with Leavitt's linear relation. A recent relationship found is

M_{V}=-2.76\log_{10}P\,-1.40

It was thus possible to obtain the absolute magnitud by measuring the pulsation period of a Cepheid variable. The apparent magnitudes of the Cepheids are measured observationally as well. With the apparent and absolute magnitudes known, we can then determine the distance to that Cepheid variable using Eq.(Star Light(1)). In this way, Cepheid variables are used as standard candles to measure extragalactic distances up to 26 Mpc.

For further distances, extremely bright objects known as supernovae are used as standard candles to measure distances up to 2200 Mpc. Further discussion on supernovae will be found in a later chapter.