Astronomy 110 PRINT Name   __________________________ Fall 2005   Section 006 Homework 8 : Nearest & Brightest Stars (Due Thursday, Nov 17, 2005)

This homework is about the properties of the nearest stars to the Sun and the brightest stars in the sky. Some of these properties are summarised in tables in Appendix 5 of the textbook (pages A-13 to A-16). "Visual brightness" refers to how bright the star looks on the sky, relative to the Sun, and "visual luminosity" refers to the actual power output of the star (energy per unit time), in units of solar luminosity. Answer the following questions. In all cases, the questions refer to stars other than the Sun.

1. Which is the most luminous star on the lists? (Call this Star 1). Alpha Cygnii (Deneb)
2. Is this the hottest star on the lists? No
3. What property of the stars do you base this on? Spectral Type
4. How many stars on the lists are hotter than Star 1? 25
5. Since hot stars are more luminous than cool stars, why is Star 1 the most luminous star on the lists?
Luminosity depends both on temperature and the size of the star (it's radius). Deneb is a supergiant (as noted by its luminosity class: Ia), which means it has a huge radius. So even though there are stars hotter than Deneb on the list, Deneb must be bigger than them.

6. Of all the stars hotter than Star 1, which is the least luminous? (Call this Star 2) Alpha Canis Majoris (Sirius)
7. What is significant about Star 2?
Sirius is the brightest star in the night sky.

8. How do you reconcile your answers to the previous two questions?
Although Deneb is much more luminous than Sirius, it is much farther away, so it does not look as bright as Sirius.

9. Imagine a big sphere centered on the Sun and containing Star 1, the most luminous star. Compare the volume of this sphere with the volume of the sphere containing all the nearby stars, i.e., the stars on the first list (i.e., calculate the ratio of the volumes.)
Deneb, the most luminous star is 3000 light-years away, the nearby stars list goes out to 15 light years. To compare the volume of a sphere from the Sun to these points we use Volume=(4/3)*pi*R3. The ratio of the volumes is then VDeneb/VNearby=30003/153=8*106. So there is 8 million times more space in a sphere around the Sun that ends at Deneb, than there is from the Sun out to 15 light-years.

10. Are there any nearby stars with the same luminosity class as Star 1? No
11. How many main sequence stars are there in the local sample? 19
12. What do your answers to the previous two questions tell you about the relative scarcity of main sequence and supergiant stars?
Deneb's luminosity class is Ia, a supergiant. Main sequence or "regular stars" like the Sun are in class V. There are several main sequence stars within 15 light-years of us, but no supergiants, so this implies that supergiants are rare.

13. Find another star that has the same temperature as Star 1: Beta Carinae (Miaplacides) or Beta Aurigae (Menkalinan)
14. Recall the relation among a star's luminosity, temperature and radius. Calculate the relative sizes of Star 1 and this star (i.e., the ratio of their radii).
Luminosity is proportional to the radius-squared times temperature-to-the-fourth-power (R2T4). This means for two stars with the same temperature we can determine the ratio of their radii from their luminosity. The formula is RDeneb/RBeta...=(LDeneb/LBeta...)1/2. Recall that taking something to the 1/2 power is the same as taking a square-root. Deneb's luminosity (relative to the Sun) is 2.7*105 and Miaplacides' is 210. Putting this into the equation tells us that Deneb's radius is about 36 times bigger than Miaplacides'. If you chose Menkalinan, whose luminosity is 94 times the Sun, you'd find that Deneb is about 54 times bigger than it.

15. Describe briefly the processes going on near the core of Star 1.
The huge amount of gravity at the center of a star pushes particles very close together. Under these conditions, lighter elements can fuse together to form heavier elements, in a process called fusion. For main sequence stars, four protons (aka hydrogen nuclei) collide and fuse together and form a helium nucleus (two of the protons convert to neutrons in the process). However, Deneb is a supergiant, which means that it has used up the hydrogen in its core and is now fusing together heavier elements.