Section 5–6 Large distances

(Nearby stars / Distant stars / Sizes of galaxies and universe)

In this section, the three interesting points discussed are the distance of nearby stars, distant stars, as well as sizes of galaxies and universe.

1. Nearby stars:

“…The distance of nearby stars can be measured by triangulation, using the diameter of the earth’s orbit as a baseline (Feynman et al., 1963, section 5.6 Large distances).”

In general, we do not always measure distance by using a meter stick. Dr. Sands explains that we can also measure the horizontal distance between two mountains using a method of triangulation. This means that we have a different definition of distance depending on the method of measurement. Importantly, the distance measured by different methods should agree within an acceptable uncertainty. For example, we measure the distance from the earth to the moon by using two telescopes at different places on the earth that give us two angles subtended. Similarly, we can focus a telescope on a star in summer and in winter to determine the two angles. Thus, the distance of nearby stars can be measured using the method of triangulation in which the diameter of the earth’s orbit is used as a baseline.

According to Percy Williams Bridgman (a Nobel laureate), “[w]e never have under observation more than two angles of a triangle, as when we measure the distance of the moon by observation from the two ends of the earth's diameter. To extend to still greater distance our measures of length, we have to make still further assumptions, such as that inferences from the Newtonian laws of mechanics are valid. The accuracy of our inferences about lengths from such measurements is not high. Astronomy is usually regarded as a science of extraordinarily high accuracy, but its accuracy is very restricted in character, namely to the measurement of angles (Bridgman, 1927, pp. 16-17).” He also suggests that the concept of length which is measured by different methods should have different names.

The method of triangulation in determining the distance of nearby stars to earth is also known as “stellar parallax method.” In 1838, Friedrich Bessel was the first to successfully measure a stellar parallax for the star (61 Cygni) by using a Fraunhofer heliometer. In other words, he determined a trigonometric parallax in which a nearby star appears to change its position with respect to the background of distant (“fixed”) stars as the Earth moves from one side of the sun to the other in half a year.

2. Distant stars:

“… If one now measures the color of a distant star, one may use the color-brightness relationship to determine the intrinsic brightness of the star (Feynman et al., 1963, section 5.6 Large distances).”

According to Dr. Sands, physicists first determine the color of a distant star and then use the color-brightness relationship to deduce the intrinsic brightness of the star. By measuring the apparent brightness of the star, they could compute the distance of the distant star that is farther away. Essentially, the apparent brightness of the star decreases with the square of the distance. The correctness of this method of measuring stellar distances was confirmed by the results obtained for groups of stars known as globular clusters. By looking at the photograph of stars in globular clusters, one might be convinced that these stars are all together. Dr. Sands elaborates that the same result can be obtained from measurements of distance by using the color-brightness method.

In 1908, Henrietta Leavitt published her findings of Cepheid variables in which their periods of brightness are related to their apparent brightness. A Cepheid variable is a type of pulsating star that is varying in both its diameter and temperature and producing changes in brightness with a well-defined period and amplitude. In 1913, Ejnar Hertzsprung realized the significance of this discovery and established an important distance measuring tool for distant stars and it is sometimes known as the standard candle method. Specifically, Cepheid variables can be used to measure distances of distant stars from about 1 kpc to 50 Mpc by using the distance modulus equation: m - M = 5 log d - 5. A parsec (or a pc) is equal to about 31 ´ 10¹⁵ m.

3. Sizes of galaxies and universe:

“…Knowing the size of our own galaxy, we have a key to the measurement of still larger distances—the distances to other galaxies (Feynman et al., 1963, section 5.6 Large distances).”

Dr. Sands explains that knowing the size of our galaxy provides a key to the measurement of even larger distances, including the distances to other galaxies. According to him, there were information and evidence to support the idea that the sizes of galaxies are all in the same order of magnitude. (This is not exactly true because galaxies can be classified as giant galaxies and dwarf galaxies.) Thus, by using the method of triangulation again, one might measure the angle subtended by a galaxy in the sky and deduce its size. By assuming its size is similar to size of our galaxy, some astronomers estimated the distance from its apparent size to be 30 million light-years from the earth.

Dr. Sands mentions that some of the galaxies are about halfway to the limit of the universe’s size (10²⁶ meters). Currently, based on an interpretation of the 7-year Wilkinson Microwave Anisotropy Probe (WMAP) data, the diameter of the observable Universe is deduced to be 28.3 ´ 10⁹ parsecs (Bielewicz & Banday, 2011). In other words, the distance between the earth and the edge of the observable universe is about 14 billion parsecs. In a sense, the distance travelled by the light from the edge of the observable universe is close to the age of the universe times the speed of light, However, this is not accurate because the edge of the observable universe and the earth are moving further apart. More important, we are unable to observe light beyond the edge of the observable universe, and thus, the size of the universe could be far larger.

Questions for discussion:

1. How do physicists measure the distance of nearby stars?

2. How do physicists measure the distance of distant stars?

3. How do physicists measure the sizes of galaxies and universe?

The moral of the lesson: the distance of stars can be measured by using the method of triangulation and the standard candle method.

References:

1. Bielewicz, P., & Banday, A. J. (2011). Constraints on the topology of the Universe derived from the 7-yr WMAP data. Monthly Notices of the Royal Astronomical Society, 412(3), 2104-2110.

2. Bridgman, P. W. (1927). The Logic of Modern Physics. New York: Macmillan.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.