29:50 Modern Astronomy
Fall 2002
Lecture 11 ...September 25, 2002
The Hertzsprung-Russell Diagram and the Mass-Luminosity Relation

(1) The Hertzsprung-Russell diagram (continued).

• Define axes
• Look at Figure 16.20 of book for diagram with stars filled in
• Main sequence, giants, supergiants, white dwarfs
• Transparency with real data from Hipparchos. Why are there no supergiants in the Hipparchos data?

(2) The Significance of the Hertzsprung-Russell diagram

• ``Dwell Time'' as stars move around in HR diagram. Regions with a lot of stars correspond to long-lived stages of stellar evolution.
• Astrophysical significance of long lifetime of Sun.

(3) The Key to Understanding the Whole Diagram is Understanding the Main Sequence

(4) The Main Sequence is a relation between T and L. What else is involved?

(5) Bald statement of fact: will justify later. The Main Sequence is a relation between the luminosity of a star and its mass. This is cleverly called the Mass-Luminosity Relation.

(6) How do we measure masses of stars?

(7) Types of binary stars:

• Visual binary stars
• Eclipsing binary stars
• Spectroscopic binary stars.
• Picture of visual binary
• Hypothetical picture of Beta Lyrae
• The Binary nature of Sirius
• Illustration of binary nature of Mizar

(8) The Nature of Gravity. You can't really get very far in astronomy without talking about gravity.

Idea of forces. Apply a force to an object and it accelerates
Demonstration.

(9) Gravity=force between pieces of matter because they have mass.

Two objects, with masses M (the big guy) and m (the little guy) separated by a distance R (in meters). The force is

where is one of the fundamental constants of nature.

(10) Idealization of ``Circular Orbit Equation''
diagram with geometry of circular orbit equation.
Question: what is relation between M,m,V,R?

(11) Introduce concept of ``centrifugal force''. For object moving in circular motion with speed V, radius of circle R, and having mass m, there is a force which must be exerted to keep it so moving.

(12) Circular Orbits: Equate two forces.

so

or

From a measurement of V and R, and knowing G, we can compute M.

Let's work an example. We observe a little (nonmassive) star orbiting a big (massive) star in a circular orbit. The radius of the orbit is 5 million kilometers. The speed of the little star in its orbit is 283 kilometers per second. What is the mass of the big fella?

We use the equation . Before plugging in numbers, we have to be picky and convert the radius to meters, and the velocity of meters per second. Soooo... 283 kilometers per second = . And R = 5 million kilometers meters. Now we are ready to plug in numbers.

kilograms. That is three times the mass of the Sun.

These concepts will show up again and again in astronomy, and are the conceptual keys for understanding ideas like black holes, dark matter, and the dark energy.

Demonstration of Cavendish experiment.

The Mass-Luminosity Relation From observations of binaries we have masses of a few hundred stars. If one plots up the masses and luminosities of stars on the Main Sequence, you find a remarkable result, termed the Mass-Luminosity Relation. It is shown in Figure 16-22 of your textbook.
Transparency of Figure 16-22.

This transparency shows that the stellar luminosity goes up extremely fast as the mass increases. If we express this in terms of a formula, we write

If we look at this diagram, we see that a star with 10 times the mass of the Sun is not merely 10 times as luminous, but instead several thousand times as luminous. We know of stars that are as much as 25 times as massive as the Sun. This relationship tells us that their luminosity would be 78,000 times that of the Sun.

The mass luminosity relation leads to an extremely important conclusion about stellar lifetimes. Stars use their own mass as their source of fuel. Since massive stars are using it up at a prodigious rate, they cannot last as long as lower mass stars.

Let's work out an illustration. Take two cars. A volkswagen has a gas tank of 10 gallons, and gets a mileage of 50 miles per gallon. A Cadillac ``Bon Vivant'' has a 20 gallon gas tank, but gets 4 miles to the gallon. Even though the ``Bon Vivant'' has twice as much fuel, it doesn't travel as far as the volkswagen because it if using up the fuel at a much faster rate.

The same kind of reasoning works for stars. A 10 solar mass star has ten times as much fuel as the Sun, but it is using it up at about 5000 times the rate of the Sun. This means that the ``lifetime'' of the star will be much less, about 1/500 the lifetime of the Sun.

This is illustrated in Figure 19.9 of your textbook.
Info on stellar lifetimes versus mass.

The mass-luminosity relation says that if we see a very massive main sequence star, it must have formed rather recently.
This result will help in understanding both where stars come from, and what they do after the Main Sequence.

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