More About Stars
Lecture 7
Stellar magnitudes:
The magnitude system originated with Hipparchus in the second century B.C. He grouped the stars into six categories by brightness. The brightest stars were assigned to the “first magnitude” group and the faintest to the “sixth magnitude group.” Stars corresponding to sixth magnitude are the faintest stars visible to the naked eye.
Now we would consider Hipparchus’ magnitudes to be apparent magnitudes – magnitudes that correspond to how bright the stars appear.
A first magnitude star is 100x brighter than a sixth magnitude star.
The human eye responds logarithmically to light, so each change of one magnitude corresponds to a factor of 2.5 in brightness. Over five magnitudes, it corresponds to (2.5)5, which is about 100.
To modernize the magnitude system, we define the scale such that a difference of five magnitudes is exactly a factor of 100 in apparent brightness. Also, the scale is extended to magnitudes smaller than 1 and larger than 6. Decimal magnitudes are also permitted.
A second magnitude can also be assigned to each star; this magnitude corresponds to the true brightness of the star (rather than to its appearance from Earth). We call this the absolute magnitude.
A star’s absolute magnitude is equal to the apparent magnitude that it would have if located at a distance of 10 pc from the observer.
If a star is exactly 10 pc away, its apparent and absolute magnitudes are equal.
If the star is closer than 10 pc, its apparent brightness increases, so its apparent magnitude becomes a smaller number (since smaller magnitudes mean brighter stars) than its absolute magnitude.
If the star is farther than 10 pc, its apparent magnitude becomes larger than its absolute magnitude.
Imagine that you are viewing a star that has an apparent magnitude of 0.2 and is located about 100 parsecs away from us. Which of the following is most likely the star’s absolute magnitude?
Star G has an apparent magnitude of 5.0 and an absolute magnitude of 4.0. Star H has an apparent magnitude of 4.0 and an absolute magnitude of 5.0. Which of the following statements is true about viewing these two stars from Earth?
Vega has an apparent magnitude of 0.03 and an absolute magnitude of 0.58. If it were moved twice as far from Earth as it is now, which of the following would occur?
Pollox has an apparent magnitude of 1.1 and an absolute magnitude of 1.1. Epsilon Eridani has an apparent magnitude of 3.72 and an absolute magnitude of 6.1. From which of these stars do we receive more light at earth?
Spectral Types of Stars
The spectra of different stars have different appearances as a result of the stars’ different temperatures. Recall that absorption lines in the visible light portion of the spectrum are created by hydrogen when the hydrogen atoms begin in the second energy state and then absorb photons. How did the atoms get to the second energy state to begin with? Some may have gotten there due to other photons, but the majority of the atoms were excited by collisions with other atoms within the stellar atmosphere. Hotter gas means more violent (and more energetic) collisions.
Stars around 10,000K have a larger fraction of hydrogen atoms in this second energy state than stars of any other temperature. Cooler stars tend to have more atoms in the ground state and hotter stars tend to have more atoms in a higher state. Therefore, the hydrogen absorption features of these stars are weaker; fewer atoms are at the necessary starting point to make use of visible light photons (the red, blue, and purple ones we saw in class). Similar conditions apply to other elements & ions, but plain old hydrogen is the easiest to understand. Different temperatures are best for different elements (in terms of allowing them to produce deep, dark absorption lines).
When stellar spectra were first collected, the above processes were not understood, so the stars were put into groups based on which elements appeared in their spectra. These spectral classes were then named based on the strength of their hydrogen absorption lines: type A, type B, etc.
In the first half of the 20th century, Cecilia Payne applied the (then) new laws of quantum mechanics to these stellar spectra for her doctoral thesis at Harvard. She concluded that the actual chemical content of the stars varied very little from one star to another. Instead, different stellar temperatures resulted in the different appearance of the spectra. In addition, she concluded that all stars are primarily made up of hydrogen.
Since then, when we refer to spectral types, we still use the groupings that were made before Cecilia Payne came along, but we think of them in the following order:
OBAFGKM
from hottest to coolest. Some groups were consolidated and/or eliminated as well.
We also use subgroups for greater specificity. These consist of a single numeral placed after the letter corresponding to the spectral type. For example, the Sun is a G2 star. A slightly cooler star might be classified as a G3 star while a slightly hotter star might be classified as a G1 star. An F9 star is hotter than a G0 star. Etc… In an attempt to split hairs even more finely, occasionally decimal spectral types turn up.
In any case, the advantage of spectral types is that the presence or absence of particular spectral lines can quite accurately determine the temperature of a star. This is an advantage because the blackbody spectrum of a star is not always easy to observe; sometimes the profusion of spectral lines obscures the overall shape, sometimes the peak is outside visible wavelengths, and sometimes it is just easier to take a simple spectrum than to accurately measure a star’s brightness.
Sizes of Stars
Ideally, we would measure the sizes of other stars in the same way that we measure the size of the Sun, by measuring the angular size of the star and using its distance from us to convert the angular size into a true size. Unfortunately for this method, most stars are far too distant & too small for this to work. With a few exceptions, most stars appear to be point-like, even in our largest telescopes.
We can determine the stellar size indirectly using Stefan’s Law:
Luminosity of star (measured from flux + distance) = area of star * flux at stellar surface
Luminosity = 4 p R2 * s T4
Luminosity is proportional to R2 T4
For example, the star Mira has half the Sun’s surface temperature (3000K rather than 5800K) and 400 times the luminosity of the Sun (1.6 x 1029 W rather than 4 x 1026 W).
So, in solar units:
L = R2 x T4 à 400 = R2 x (1/2)4
à 400 = R2 x (1/16)
à 6400 = R2
à R = 80 solar radii
Another example: the star Sirius B has four times the Sun’s surface temperature (24,000K rather than 5800K) and 0.04 times the luminosity of the Sun (1025 W rather than 4 x 1026 W).
So, in solar units:
L = R2 x T4 à 0.04 = R2 x (4)4
à 0.04 = R2 x (256)
à 0.00015625 = R2
à R = 0.0125 solar radii
Hertzsprung-Russell Diagram
This very useful diagram is used routinely by astronomers to display the properties of stars. It is a graph on which the temperature (or spectral type) of a star (or group of stars) is plotted on the x-axis (and increases towards the left) and the luminosity of the star/stars is plotted on the y-axis. Interestingly, stars turn out to occupy only certain parts of such a diagram. The majority of the stars fall along the main sequence, which runs from the lower right of the diagram towards the upper left. Another group of stars is seen in the upper right, the “red giant” region, and a third group is seen in the lower left, the “white dwarf” region. These names refer to the colors & sizes of these stars. Notice that in order to plot a star on the HR Diagram, we need to know its luminosity, which requires knowing its distance.