Star Age as a Current Scientific Clock

By Kevin Mellem

The stars in the heavens have caused observers to ponder many different questions over the centuries. Where did they come from? How did they come to be? How old are they? Fortunately, astronomical science has advanced to the point where we can now provide an answer to this last question. By measuring both the luminosity of a star, which is its brightness relative to the brightness of our sun, and the color spectrum of the star, which represents the surface temperature of the star and the rate at which it is burning its nuclear fuel, scientists are now able to estimate the age of the stars of the universe (Dalrymple 2004).

Luminosity and effective temperature are closely related to one another, as both characteristics are a consequence of the forces at work within a star. Stars originate when a cloud of gas collapses in upon itself due to gravitational forces. This collapse of the gas cloud releases large amounts of gravitational energy which in turn produces an enormous internal temperature. This large internal temperature is enough to cause the hydrogen gas within the cloud to burn via nuclear fusion reactions that produce helium. These nuclear fusion reactions generate a tremendous amount of heat and light, thereby providing the fuel for the star's luminosity and high temperature for the majority of its life (Dalrymple 2004).

Luminosity can be defined as the total radiation emitted per second in all directions by a star, and it is expressed in units of solar luminosity, where our sun has a luminosity value of one. In order to find the luminosity of a star, the distance from the earth to the star must be known. This distance can be calculated for the closest stars using two measurements and some simple algebra. To begin, an observer measures the position of a star against the background of more distant stars at two times that are six months apart; the observer can then calculate the apparent shift of the star in the sky, known as the parallax of the star. This information can be used, along with the known diameter of the earth's orbit, to find the distance from the earth to the star according to the following equation:

Star Distance = (Diameter of Earth's Orbit) / (Parallax of Star)

By combining this information with the measured radiation emitted by a star, scientists are able to use the principle that the apparent brightness of a star is diminished by the square of its distance to calculate the star's stellar luminosity. Stellar luminosity values can be as small as 0.1 times the luminosity of our sun to as large as 60,000 times as luminous as our sun (Motz 1975).

The surface temperature of a star is much easier to determine, as it can be evaluated using the observable color of the star. The color of a star is directly related to the wavelength of the radiation emitted by that star. Physicist Max Planck developed the following equation that directly related the radiation emitted by any hot body to the temperature of that body, where I(ν) is energy per unit time per unit surface area per unit solid angle per unit frequency, h is Planck's constant, ν is the frequency, c is the speed of light, c/ν is the wavelength of the light, k is Boltzmann's constant, and T is the temperature of the black body:

I(v)=((2hv^3)/c^2)*(1/((e^(hv/kT)-1)) (1)

Equation information from "Planck's law of black body radiation," Wikipedia, http://en.wikipedia.org/wiki/Planck%27s_law_of_black_body_radiation, Last Updated: 12/16/05. Accessed 12/16/05.

Using this equation, scientists have calculated the surface temperatures for many stars, with values ranging from 3,000 K for the colder red stars to 40,000 K for the hottest blue-white stars (Motz 1975).

In reality, however, the age of any given star is not estimated simply by evaluation of its own luminosity and surface temperature values. In practice, the age of a star is evaluated by determining an age for the globular cluster of stars to which the star belongs. A globular cluster of stars is a group of gravitationally attracted stars that can range in number from tens of stars to more than a million stars per cluster. Each cluster of stars is unique, and all of the stars of a cluster were formed at the same time. As a consequence of this simultaneous formation, all stars within an individual cluster are the same age (Bolte 1998). It should also be noted that the luminosity and surface temperature of a star are only used by scientists to determine the lifespan of a star - they do not define the actual age of a star. The lifespan of a star is simply a function of how much nuclear fuel it contains and the rate at which it is burning that fuel. Larger, hotter stars contain more fuel than smaller, colder stars, but they are burning their fuel at much higher rates than the smaller stars. As such, the larger stars burn out more quickly and exist for a shorter period of time than smaller stars (Dalrymple 2004). This burning of nuclear fuel defines the lifespan and therefore the stellar evolution of a star, which we can observe using the luminosity and surface temperature of the star.

In reality, however, the age of any given star is not estimated simply by evaluation of its own luminosity and surface temperature values. In practice, the age of a star is evaluated by determining an age for the globular cluster of stars to which the star belongs. A globular cluster of stars is a group of gravitationally attracted stars that can range in number from tens of stars to more than a million stars per cluster. Each cluster of stars is unique, and all of the stars of a cluster were formed at the same time. As a consequence of this simultaneous formation, all stars within an individual cluster are the same age (Bolte 1998). It should also be noted that the luminosity and surface temperature of a star are only used by scientists to determine the lifespan of a star - they do not define the actual age of a star. The lifespan of a star is simply a function of how much nuclear fuel it contains and the rate at which it is burning that fuel. Larger, hotter stars contain more fuel than smaller, colder stars, but they are burning their fuel at much higher rates than the smaller stars. As such, the larger stars burn out more quickly and exist for a shorter period of time than smaller stars (Dalrymple 2004). This burning of nuclear fuel defines the lifespan and therefore the stellar evolution of a star, which we can observe using the luminosity and surface temperature of the star.

Using these luminosity and surface temperature values of the stars within a single cluster, scientists are able to establish an age of the cluster (and thus the stars within the cluster) using a Hertzsprung-Russell (H-R) diagram. In an H-R diagram, luminosity is plotted as a function of temperature, as shown in Fig. 1 below:

Figure 1: A Hertzsprung-Russell Diagram of Globular Cluster M5

figure1 star luminosity

H-R diagram obtained from "The Hertzsprung-Russell Diagram of a Globular Cluster," http://www.phys.unsw.edu.au/astro/wwwlabs/gcCm/gcCm_intro.html. Accessed 12/18/05.

As is evident in Fig. 2, the majority of stars in a cluster lie along a continuous line that slopes diagonally from the upper left portion of the diagram to the lower right portion of the diagram, which is known as the main sequence (labeled MS in the diagram). In this main sequence, the upper right contingent of the sequence is composed of the most massive, hottest, and brightest burning stars, while the lower right portion is comprised of less massive, cooler, and least luminous stars. All stars within the main sequence rely on the nuclear fusion of hydrogen gas as the fuel for their light and heat production (Dalrymple 2004).

The portion of the H-R diagram that is used to determine the age of the star cluster is the turnoff point, marked TO in the above diagram. As a star grows older, it will eventually exhaust all of its hydrogen fuel, resulting in a dramatic expansion in size coupled with a decrease in luminosity to become a red giant. As this transformation occurs, the star will leave the main sequence and move toward the upper right corner of the H-R diagram. This stellar evolution is represented in Fig. 23 below, which diagrams the changes that occur in an H-R diagram of a cluster over time.

Figure 2: An H-R Diagram Highlighting Turnoff from the Main Sequence

figure2 main sequence turnoff

H-R diagram schematic obtained from "The Grouping of Stars in the H-R Diagram," www.ifa.hawaii.edu/users/lin/ast110-6/ast110-18.ppt. Accessed 12/18/05.

At the genesis of a star cluster, all stars fall upon the main sequence (shown in the first chart of Fig. 2 - yellow line inserted for reference to initial condition of diagram). However, the large, bright stars of the upper main sequence burn their fuel more quickly and transform into red giants sooner than the smaller stars of the lower main sequence. Thus, in a younger star cluster, the turnoff point will form at the top of the main sequence, where the more massive stars are beginning to become red giants (represented in the second chart of Fig. 2). As a cluster ages, less massive stars will deplete their hydrogen fuel, and the turnoff point will move down along the main sequence. Eventually, in the oldest clusters, the turnoff point will be near the bottom right of the main sequence (chart 3 of Fig. 2). This migration of the turnoff point over time is what allows scientists to determine the age of a star cluster - they simply need to determine the mass of the stars leaving the main sequence to predict the age of the whole cluster. Because scientists know the initial condition of any cluster (all stars reside on the main sequence) as well as the rate at which the stars of the cluster will evolve into red giants and move the turning point down the main sequence (determined using volume of nuclear fuel of the stars and the rate at which the stars are burning that fuel), scientists can use the movement of the turnoff point as a clock to monitor the passage of time within a cluster (Dalrymple 2004).

In addition to providing the age of the stars inside the cluster, the age of globular clusters given by H-R diagrams can be used to estimate the age of the galaxy they inhabit. Since globular clusters reside near the center of a galaxy, their age is representative of the age of the galaxy, as they must have formed at nearly the same time.

By plotting the luminosity and temperature values of stars on H-R diagrams, scientists have found the ages of globular clusters of stars within the Milky Way galaxy to be in the range of 11.5 to 14.0 billion years, and the age of the galaxy must therefore fall within this range as well (Dalrymple 2004).

Another method of dating globular clusters of stars involves measuring the age of white dwarf stars residing in a cluster. After a certain period of time, a red giant star will expel its outermost material, creating a planetary nebula and leaving the hot core of the giant in its wake. This hot core of the star is known as a white dwarf. White dwarf stars are extremely dense, and they do not utilize nuclear reactions to generate light and heat. Instead, they run on residual heat from the previous stage of their existence (Dalrymple 2004).

Since they are not producing any new heat, white dwarfs cool over time until they effectively die. The temperature of a white dwarf can be measured using its brightness, and the amount of light produced by a dwarf will decrease as the temperature decreases. Also, the rate of cooling of a white dwarf slows as the temperature diminishes - a factor that must be accounted for when dating a white dwarf. Using the known initial condition of the dwarf, its current brightness (and therefore its temperature), and the rate of cooling, one can extrapolate the data to determine the age of a white dwarf. Scientists have utilized the Hubble Space Telescope to observe white dwarfs in the M4 globular cluster of the Milky Way galaxy, and their observations indicate that the white dwarfs have an age of 12.7 ± 0.7 billion years, which can also be taken as an approximate age of the galaxy. This age is in good agreement with the age given by the H-R diagram method, and the agreement between the two methods suggests that the age of the stars and the Milky Way galaxy is greater than 11 billion years (Dalrymple 2004).

Sources:
  1. Bolte, Michael. 1998. "Measuring Stellar Ages." http://www.ucolick.org/~bolte/AY4/notes9/node2.html. Accessed: 12/16/05.
  2. Dalrymple, G.B. 2004. Ancient Earth, Ancient Skies. Stanford University Press, Stanford, Ca.
  3. Motz, Lloyd. 1975. The Universe: Its Beginning and End. Charles Scribner's Sons, New York.