## Basic Stellar Properties

You will derive the mass and luminosity of the sun in your homework, but for now, let's just start out with some basic properties.

For simplicity, we assume the sun to be pure hydrogen.

The mean density of the sun found from its mass (2E33 grams) and its radius (7E10 cm) is 1.4 grams cm-3.

The mean temperature gradient in the sun is approximately equal to the core temperature divided by the radius. The core temperature of the sun is approximately 15 million degrees which leads to a temperature gradient on the order of 10-2 K/meter.

This will become important later when we discuss the mean free path of photons in the solar interior.

Next we will demonstrate how integrating the planck curve shows that the total energy of a star depends on the fourth power of its blackbody temperature.

You integrate the planck function over all wavelengths, make a variable substitution to produce an integral that you can look up in a book and voila: The planck function is an accurate description of the radiation given off by a dense object as long as thermodynamic equilibrium holds.

Under TE, the temperature of the matter and radiation are the same. The only way this can occur is if the radiation shares energy with the matter. This condition occurs whenever the radiation can not travel very far through matter without being absorbed or scattered. This condition is known as opacity

If you consider radiation passing through a slab or layer of material, the amount that radiation interacts with the material depends on three things:

• The total thickness of the layer (i.e. the path length through the material).

• The density of atoms in the layer

• The absorption coefficient or cross-section of the atoms (which depends on chemical composition, degree of ionization, etc). This is usually denoted by the greek symbol kappa (k) and its frequency dependent.
We can write the reduction in intensity, dI, as a function of these as follows:

dI = -krIdz
Total absorption is set by the condition that dI/I = 1.

For a layer at about 1/2 a solar radius, k = 1 (in cgs units) and we can take the density at this location to be the same as the mean density of the sun, which is about 1 as well.

Hence, at this layer, the mean free path of radiation is only 1 cm.

Measuring Stellar Masses

Newton's generalization of Kepler's third law (derived below) can be used to measure the masses of stars in binary systems. Observations of the mutual orbit then yield values for a and P (the orbital period). While these can be messy for close binaries or for binaries that are not well aligned along the line of site, there are enough decent binary systems nearby where secure measures of stellar mass can be obtained.

These observations form the basis for the empirical mass-luminosity relationship for hydrogen burning stars.

This relationship is shown below and it reveals the important empirical scaling between Mass and Luminosity. For simplicity, let's take the scaling to be L goes as M4.

What implication does this have for stellar lifetimes?

Well the lifetime of a star is equal to its total fuel (which is M), divided by the fuel consumption rate, which is the Luminosity of the star).

so lifetime goes as M/L which goes as M/M4 or M-3.

The calibrated relationship is, when M is measured in solar masses, the lifetime is:

1010 years / M3

Thus a 10 solar mass star would have a lifetime of only 10 million years. This provides a useful way for age dating stellar clusters: Stellar Clusters are useful ``laboratories'' for testing our theories of star formation.

Laboratories are generally thought of as places where scientists can run controlled experiments to test their hypotheses and theories. The galaxy has helped us out to some extent by creating stars in clusters, instead of creating them one by one in random places, at random times, under wildly varying conditions of temperature, density, and chemical composition.

Stars in a cluster formed at the same time, in the same molecular cloud.
Therefore, stars in a cluster

• have the same age,
• had the same initial chemical composition,
• and are at roughly the same distance from Earth.

Thus, when stars form within a cluster, they differ only in their mass. The more massive stars evolve more rapidly, so to find the AGE of a cluster of stars, we need merely determine the mass of the stars which have just now exhausted the hydrogen in their cores and are turning into red giants.

For instance, look at the three Hertzsprung-Russell diagrams shown below, derived from mathematical models of stellar evolution.

The first diagram is of a cluster which is only 1 million years old. The cool K & M stars have not yet settled down onto the main sequence; they are still contracting protostars, and have not yet ignited hydrogen fusion in their cores. On the other hand, the hottest O star has already been converted to a red supergiant. . This would be about a 50 solar mass star.

The next diagram shows the cluster at an age of 10 million years. Here the main sequence turnoff corresponds to about a 10 solar mass star. The The next frame is of the cluster that is 100 million years old. The main sequence lifetime of a 6 solar mass star is 100 million years, so stars with M = 6 Msun (L = 530 Lsun, spectral type A) are just turning off the main sequence. The next frame shows the cluster at an age of 1 billion years, which corresponds to the lifetime of a 2 solar mass star (spectral type F). By this age, a few of the more massive stars have no evolved to populate the white dwarf sequence. The final diagram is of a cluster which is 10 billion years old. The main sequence lifetime of a 1 solar mass star is 10 billion years, so stars with M = 1 Msun (L = 1 Lsun, spectral type G) are just turning off the main sequence. All the stars on the red giant branch are more massive than 1 solar mass. More white dwarf stars are also present. Fortunately, the mathematical models provide a good fit to the Hertzsprung-Russell (H-R) diagrams which are actually observed for clusters of stars.