Breaking the Celestial Sphere

The nearest stars are much further away than the most distant object with our solar system. So placing them all on a fixed sphere a long way away worked well. We measure how the solar system objects move relative to the fixed background of stars.

When we expand our investigations beyond the solar system, one of the first things we have to realize is that our construction of the celestial sphere now loses much of its usefulness.

Now we must realize two things about stars:

1) Stars are not all the same distance away.

Figuring out the distance to stars is a major project of astronomy.

2) Stars are not fixed. They are moving. (with respect to our Sun.)

They are far enough away that this motion is very difficult to detect.

Star’s Motion

Although we tend to think of stars embedded in a “fixed background”, they really are not.

All stars have some motion relative to the Earth.

To determine the velocity of a star we need to know its radial velocity and its tangential velocity.

Radial velocity is velocity directly toward or away from the Earth. We know we can determine this by measuring the doppler shift of the absorption spectra of the stars.

To determine tangential velocity we need to measure the proper motion of a star and its distance from the sun.

Proper motion is the angle that a star moves through divided by the time its takes the star to move through that angle.

Bernard’s star has a proper motion of 10.25 arcsec per year! This is the star with the greatest proper motion!

Tangential velocity then is the proper motion times the distance the star is from Earth.

These two stars have the same proper motion. Which one is moving faster?

The total velocity is the combination of the radial velocity with the tangential velocity.

tangential

velocity

Notice that we need to know the distance to the star to properly determine its velocity.

What do we know about the nature of Stars?

Stars are defined to be “self-luminous celestial objects”, that is objects in the sky that produce their own light. The obvious star in our sky is the sun, which dominates our view of the universe.

We have already learned what the source of the sun’s energy is (nuclear fusion reactions) and we can safely assume that this process fuels most other stars as well.

We have found by understand the nature of light and the atomic nature of matter that we can determine temperature, chemical composition and even radial velocity of objects by carefully observing their spectra.

However, the power or luminosity, the size, the mass and the motions of stars are very difficult to determine.

All of these properties can be determined very quickly if we know the distance to the star.

Stellar Parallax

We know what stellar parallax is and we know that it must be hard to measure, since Tycho Brahe with his accurate measurements could not detect it.

However, with the use of modern telescopes stellar parallax can be measured and for near-by stars we can use this method to determine the distance to these stars.

Using a little geometry (as shown in figure 13-2 in the text) knowing the parallax angle allows an easy determination of the distance.

If we define a parsec as the distance to a star whose parallax angle is 1 sec of arc (1/3600 of a degree). The equation for the distance is simply

Distance in parsecs = 1 / (parallax angle in arc sec)

Defined like this, 1 parsec = 3.26 light-years.

The nearest star to the Sun makes a parallax angle of 0.77 arc sec. How far away is the star in parsecs? How far in light years?

Stellar parallax is one of the very few ways we have to directly measure the distance to stars.

Astronomers have been able to measure the distance to over 1 million stars using stellar parallax.

Stellar parallax is really the first way we have to begin to judge the size of the universe. But we need ways to find distance to objects whose parallax angle is too small to measure even with the best telescopes.

Apparent brightness and Luminosity

A property of any star that can easily be measured is its apparent brightness. This is just a measure of how bright the star appears as viewed from Earth. It is equivalent to the intensity of the starlight. (recall intensity is energy per second per square meter.)

Another property of a star that is not so easy to measure its Luminosity. Luminosity is the total amount of energy the star produces every second. This is the power or wattage of the star.

There are two reasons why two stars would have different apparent brightness.

1) They have different Luminosities.

2) They are different distances from Earth.

This is much like the size of the Sun and Moon, both have about the same angular size. Are they big and far away or small and close-up?

Similarly, stars of the same apparent magnitude could have high luminosity and be far away or low luminosity and close up.

Inverse Square law

However, we can figure out how much dimmer stars will appear as they get further away.

We can imagine ourselves at different distances from the star and calculating the intensity of light we see. (Much like we calculated the total power of the sun!)

Using the spheres surrounding the star we come with a law for the intensity of light relating to the power of the star and the distance away.

Inverse Square Law º The intensity of light decreases as the inverse square of the distance from the source.

That is, if you double your distance from the source the intensity is one-fourth what its initial value. If you triple you distance the intensity drops to one-ninth its original value. (Like Newton’s gravitational law!)

In the same way we calculated the power of the Sun, we see we have a relationship between three variables: Intensity (apparent brightness), Power (Luminosity or Wattage) and Distance from Earth.

We can always measure intensity (apparent brightness), and if we know the power we can find the distance and if we know the distance we can find the power.

For any star that whose parallax angle we can measure. We not only know its distance away, but also how much total light it is producing!

However, for any star whose parallax angle is too small to measure, we don’t know the luminosity or distance of a star.

We use another property that we can measure about any star: the electromagnetic spectrum of the star.

Spectral Class

Recall that by measure the peak wavelength of a star we can determine the temperature of the star.

Further, looking at the emission/absorption spectrum of the star can tell us to what level the atoms surrounding the star are excited. This is an independent verification of the Temperature of the star.

Stars have been grouped by these two observations into a number of different spectral classes. In order of decreasing temperature the spectral classes are: O B A F G K M.

Each class is divided into 10 subgroups, using the same methods. Our sun, for example, is a G2 star. Sirius is an A1 star. Which is hotter?

Procyon A is an F5. Is it hotter than Sirius?

Capella is a binary star made up of a G0 star and a G8 star, which is hotter?

Hertzsprung-Russel Diagrams

A Hertzsprung-Russel diagram (or H-R diagram) is a very simple idea, but probably the most important idea in our study of stars.

An H-R diagram takes the set of stars whose distances we can measure by parallax, and uses this distance along with apparent magnitude to calculate absolute magnitude. The diagram plots a star’s Luminosity vs. Spectral class (temperature).

If there were no relationship between a star’s temperature and absolute magnitude the stars would be plotted all over the chart in a random way.

As it is we can see a definite pattern on the diagram that suggests a strong relationship between luminosity and temperature.

Luminosity Classes

The main feature on the H-R diagram are the stars that lie on a ‘path’ moving from low temperature/ low wattage stars to high temperature/high wattage stars.

These stars are called main sequence stars.

There are stars, however, that are clearly off the main sequence and are said to have a different luminosity class. The luminosity class can be determined from the star’s location on the H-R diagram, but it can also be determined by a careful examination of the star’s absorption spectra.

Other luminosity classes (besides main sequence) include giants, super giants and white dwarves.

Giants and supergiants are low temperature/high wattage stars.

White dwarves are high temperature/low wattage stars.

What do you think is different about these stars from main sequence stars?

Spectroscopic parallax

Spectroscopic parallax is a method for measuring the distance to far away (parallax angle is too small to measure) stars by measuring the spectrum of the star and using the H-R diagram.

By measuring the spectrum of the star, astronomers are able to determine the spectral class and luminosity class of a star.

Knowing these two properties of the star we can go to the H-R diagram and see what absolute magnitude the spectral class of the star corresponds to (within an a small range).

When we know the absolute magnitude, then using its apparent magnitude we can determine the distance to the star.

Star’s spectra gives

Spectral/luminosity classes gives (via H-R diagram)

Absolute magnitude (w/ apparent magnitude) gives

DISTANCE !

So we now have a method for getting a good approximation of the distance to stars where stellar parallax can’t be used.

Note that this method depends on believing that stars close to the Earth are representative of stars across the universe. That is, the Earth is not at a special place in the universe.

What a change this idea is from the old geocentric model, which had the Earth at a very special place: the center of the entire universe.

Star Sizes

One might think that the temperature of an object would tell you exactly how much light an object is emitting so that measure the temperature of a star is the same as measuring its absolute magnitude.

(The main sequence suggests that there is a relationship there but, also, that there must be another factor involved.)

Imagine a 100 W light bulb by itself will emit a certain amount of light at a certain temp. 1000 light bulbs will glow at the same temperature but emit 1000x more light.

That is, The temperature of a star will tell us how much light is emitted per surface area.

It is the temperature and amount of total surface area that determine brightness.

So white dwarves are hot, but very small, so their luminosity is low.

Giants and supergiants are cool (for stars) but very, very large, so their luminosity is high.

Even in the main sequences star sizes range from one-tenth the sun’s size to 20x the sun’s size, which affects their luminosity.

Sizes of stars can also be determined from H-R diagrams and spectra.

Spectra tell us a star’s temp and thus the amount of light they emit per surface area.

The H-R diagram is used to determine total luminosity.

These two factors are used to find surface area and thus the diameter of the star

Temperature gives H-R diagram gives

Light per unit area Total Amount of Light

gives

Total Surface Area

and

Diameter