1. The Family of Stars

2. Properties of Stars
We have already determined how to measure distances to stars (parallax) and the temperatures of stars. Now we will focus on:
1. How much energy they emit
2. How big they are
3. How much mass they contain

3. Quick Review
Parallax equation: d = 1/p where d is distance in parsecs (pc) and p is the parallax angle in arcseconds ("). Through the use of parallax, most stars are extremely far away.
Parallax is often measured when the Earth sweeps out a large angle in its orbit. Fig
The Stefan-Boltzmann Law describes the energy output of a star in terms of its temperature in K. E = T4

4. William Herchel
Found 1st magnitude 100 times brighter than 6th magnitude
5 intervals /5 = 2.512
One magnitude difference corresponds to a factor of times brighter/dimmer
Difference in brightness between 1st magnitude &
2nd = 2.5
3rd = 2.5 x 2.5 = 6.25
4th = 2.5 x 2.5 x 2.5 = 15.6

5. STANDARDS Deneb is a standard star of magnitude =1
Need standards to determine other magnitude
Venus is 100 times brighter at its brightest point therefore magnitude of 1-5 = -4
Suns m= -26.5
Moon= -12.6
Brighter objects have smaller magnitudes than fainter objects

6. Apparent magnitude range -27 to 28
Approximately 50 magnitudes
Every 5 there is a factor of 100 so (100)10 = 1020 factors between the faintest to brightest
apparent visual magnitude mv

7. Comparing Brightnesses
Remember - the smaller the mag. the brighter.
Two identical flashlights might appear to have different brightnesses if located at different distances. The same is true of stars!
Astronomers compare how bright a star would appear to be at a standard distance—10 parsecs. The apparent visual brightness (mv) at this distance is called its absolute visual magnitude (Mv). (not UV, IR, etc.) It can be thought of as a star’s intrinsic brightness.

8. Rethinking Bright Stars
Our sun blazes at an apparent magnitude of –26.8, but if it were moved out to a distance of 10 pc, then it would only appear to be magnitude This would place it among the dimmest stars we can see without a telescope!!
Sirius would go from –1.47 to +1.4 mags
Vega would go from to 0.5
Betelgeuse would go from to –5.6

9. Relating Brightness to Distance
Astronomers use the magnitude-distance formula to compare brightness and distance: mv – Mv = (log10[d])
If given any 2 of the 3 variables, this important equation allows us to solve for the third! Remember, smaller # = brighter!
It is often helpful to rewrite this equation as
d is in pc
mv – Mv is known as the distance modulus, a measure of distance.
The larger the modulus, the more distant the star is.

10. A Modular Example Betelgeuse: mv = +0.41, Mv = -5.6
Find its distance from Earth.
Its modulus is 0.41 – (-5.6) = 6.0
So d = 10( )/5 or 1011/5
Betelgeuse is located or 158 pc away (or 517 ly).

11. From Brightness to Luminosity
The luminosity (L) of a star is the total amount of energy it radiates in 1 second, all EM types, not just visible light.
This is different from E in Stefan-Boltzmann in that L describes the entire star, not just 1 square meter.
Astronomers start with Mv (visible only), but need to correct for the other EM wavelengths. This is small for medium temp. stars like the sun, but can be large for hot or cool stars.

12. Brightness for All Wavelengths
Adding the proper correction to Mv changes it into the absolute bolometric magnitude (Mbol), the absolute magnitude if we could see all wavelengths.
The sun’s Mbol is Other stars are compared with the sun’s luminosity, as multiples of Lsun.
Arcturus has Mbol of –0.3, so there is a difference of about 5 magnitudes, which corresponds to Arcturus being 100 times brighter, or 100Lsun.
The sun’s luminosity: Lsun = 3.83 x 1026 J/s.

13. The L Equation
Since the sun is a sphere (surface area of a sphere = 4R2 where R = radius), we can write the following equation for L:
L = (4R2)(T4) [J/s]
If we compare another star to the sun, then we get (many constants cancel):
All these terms are now ratios.

14. Astronomers’ Periodic Table!
Classes of stars can be sorted out and displayed in an important diagram that was named after its creators: the Hertz-sprung-Russell (HR) Diagram.
On its vertical axis is luminosity (L/Lsun) or absolute magnitude, since they are related.
On its horizontal axis is temperature or spectral type, since they are again related.
Stellar size (as a ratio to the sun) can be shown along diagonal lines.

15. The HR Diagram

16. Another look a the HR Diagram

17. Families of Stars
The HR Diagrams separate families of stars into different regions of the diagram.
The main sequence (V) runs from the upper left to the lower right. 90% of stars fall into this category, including the sun.
Giant (II or III) stars lie right above the main sequence and are times larger than the sun. Some are red (cool, right), while others are blue (hot, left). Ex. Arcturus
**The Roman numerals refer to L classes. See table on p. 353 and Fig on same page.

18. More Families
Supergiants (Ia, Ib) lie at the very top of the diagram and are times the size of the sun. Some are red and some are blue. Ex. Betelgeuse is a red supergiant.
IV are subgiants (not important?)
White dwarfs are small, bright stars that are located in the lower left corner. White dwarfs are not part of the luminosity classes because they have peculiar spectra. Ex. Sirius B

19. Last Family Standing
Red dwarfs are members of the main sequence, but located in the lower right corner. They are both small and cool.
Different classes of stars will show different widths of certain lines, such as the Balmer series, due to atmospheric density. Main sequence have wide lines (most dense); giant stars are narrower; and supergiants are the most narrow (least dense).

20. Another Way to Measure Distance
Spectroscopic parallax is a way of estimating a star’s distance using:
Its spectrum  spectral class, gives the horizontal coordinate on HR Diagram
Width of spectral lines  luminosity class, gives the vertical coordinate on HR Diagram
Both coordinates on HR  its Mv (types of stars have characteristic Mv’s!
Using its mv and Mv, we can calculate the star’s distance using the distance modulus.

21. Finding Mass
More than half of all stars are members of binary star systems. Ex. Sirius, Alcor/Mizar
If binary stars are watched long enough, one can determine their orbital periods (P) and average distance between them (a). Then we can apply Newton’s version of K3:
Example on p. 348

22. Another Mass Relationship
For main sequence stars, the more massive a star, the more luminous it is. This idea is expressed in the mass-luminosity relation:
L = M3.5 (L and M are multiples of Lsun and Msun)
A star that has 10 times the sun’s mass will be about 3160 times the sun’s luminosity!