Astronomy 201  Fall'07  Problem Set 6

The four(+) problems below cover material in Chapters 15 & the first part of Chapter 16.  These should be worked out and turned in on Thursday, October 25th -- in class (preferably) but not later than 5pm.

The basic skills and definitions quiz for Chapter 16 is posted on MasteringAstronomy.com and should be taken (as assignment 6) no later than 11pm on the 25th.

Note that too many people are coming up to me after some classes asking questions about their answers being right or wrong to one among several problems.  This is not good for two reasons: (a) I have to pack up and leave the classroom promptly when the lecture is finished and (b) I don't have memorized the final answers and need to look at the solution technique to tell if someone is on the right track.  Better to ask such questions before class, so I will make myself available in my office a half hour before class --and come in ten minutes before start of class-- to answer any "last-minute" questions.  Again, it is far better to make the 7pm problem recitation class with Greg Brunner the night before the problems are due since he reviews my solutions and has time to work with you in learning how to solve the problems.

6-1: A star in the sky 100 parsecs(pc) away has an apparent magnitude of  m = +6.50.  What will be its absolute magnitude M?  (use the inverse square law of light and remember that M = m when the star is 10 pc distant)

6-2: The Sun has an absolute magnitude of M = +4.6 and is of spectral type G2V.  The star Capella , which has the same spectral type as the Sun, has an apparent magnitude of m = -0.4.

(a) What is the distance to Capella in parsecs and light years assuming its luminosity class is the same as the Sun…namely G2V?

(b) However, close analysis of its spectrum indicates that its luminosity class is that of a giant, namely G2 III.  The HR diagram indicates that such a star would have an absolute magnitude about -0.4 (i.e., M = -0.4).  If this is true, what would be the distance of Capella now, and its luminosity compared to the Sun?  [give the ratio that the luminosity is higher compare to L(sun)]

(c) Now, given the case of Capella being a giant star of spectral type G2III and M = -0.4, estimate the relative size of Capella compared to the Sun (assuming that the surface temperatures are the same).

6-3:  Supposed the Sun (G2V) and the star Sirius (A1V)  (the two brightest stars in the sky!) were in a gravitationally bound binary star system where the orbits were circular with an average separation of 10 AU and a period of 18.25 years. 

(a) What is the mass of Sirius in solar masses?

(b) Estimate the distance of the binary in parsecs and AU's assuming that the absolute magnitude of a A1V star is M = +0.6 but the apparent magnitude of Sirius is m = -1.4.

(this can be done using the equations in MI 15.1 and 15.3 or with the distance modulus equation below given as bonus problem 6-5)

(c) For the distance you get in (b), what would be the angular separation of the two stars (given their true separation is 10AU) in arc seconds seen through a telescope at the Earth? (skinny triangle again!)

6-4: (this relates to MI 16.1 material and is problem 49 in Chapter 16 of CP) Masses of the first stars.  Models of the first star-forming clouds indicate that they had a temperature of roughly 200 K and a particle density of roughly 300,000 particles per cubic centimeter at the time they started trapping their internal thermal energy. 

(a) Estimate the mass at which thermal pressure balances gravity for these values of pressure and temperature .

(b) How does that mass compare the Sun's mass?

(c) What is the estimated lifetime of a star with that mass?

BONUS PROBLEM (worth 0 or 2 points added to your problem total (0=wrong 2=right)…this is a formula derivation that you need to show every assumption and mathematical step!)

IF:  (a) m is the apparent magnitude of a star at some distance d(pc) in parsecs, and

      (b) M is the absolute magnitude of a star (equal to the apparent magnitude of the star at a distance of 10 parsecs),

      (c)  then the difference in the apparent and absolute magnitudes of a star: m - M (called the distance modulus) is related to distance by the equation:

                                                m - M = 5log[d(pc)] - 5.

            [log is the base10 logarithm; use the inverse square law of light (MI15.1) and the definitions of magnitudes (MI15.3) in this formula derivation]