Galaxies

Problem Set 2

Due in class Wednesday 8 October 2003

**The IMF &c**The Salpeter IMF is

*dN/dM = A M*^{-a}with

*a*= 2.35 from*M*= 0.1 to_{low}*M*= 120_{up}*M*_{o}.- A. Suppose a star forming event makes 1,658 stars with this
IMF. What is the normalization
*A*for this event? - B. What is the total mass of this star cluster?
- C. How does the mass change if we change
*M*from 0.1 to 0.08_{low}*M*_{o}?Assume that main sequence stars obey a mass-luminosity relation of the form

*L = M*^{3.5}in solar units. Ignore other branches of the HR diagram.

- D. What is the total luminosity of the zero age cluster?

How does this change is we change*M*from 0.1 to 0.08_{low}*M*_{o}?

If we change*M*from 120 to 100_{up}*M*_{o}? - E. Using the supplied information and assuming a one solar mass star
will live 10 Gyr, at what mass will the main-sequence turn-off
be located after 1 Gyr?

What will the luminosity of the cluster be at that age?

What will be the turn off mass and integrated luminosity at 10 Gyr?The Salpeter IMF is often used as a fiducial by which other IMF's are measured.

- F. For the Kroupa IMF given in class (with upper IMF slope
*a*= 2.35), determine the factor_{3}*X*in*M*(Kroupa) =*X*M*(Salpeter)when you integrate up the two IMFs, normalized so that both produce the same number of stars.

It is OK that*M*and_{up}*M*differ for the two IMFs - that is part of what makes this interesting._{low}

- A. Suppose a star forming event makes 1,658 stars with this
IMF. What is the normalization
**Binney & Merrifield 7.6****Binney & Merrifield 7.7****The Tully-Fisher distance to the Ursa Major cluster**Use the data of Verheijen (table 5 of his thesis chapter 5

^{*}) to determine the distance to the UMa cluster of galaxies.Do this for both the

*I*-band and*K'*-band^{**}data, for the calibration of Pierce & Tully given by Binney & Merrifield and for the HST Key Project calibration given by Sakai et al.

Be sure to include a plot of the data with calibration lines overlaid.Assuming the recession velocity of the Ursa Major cluster is 1088 km/s, what Hubble constant(s) do you find? How does this compare to the value given by Sakai et al.?

^{*}Note that in this table, Verheijen gives absolute magnitudes for a presumed distance of 15.5 Mpc.^{**}For the*K'*-band data, use the*H*-band calibrations assuming*H-K*= 0.25.