The Salpeter IMF is

with a = 2.35 from M_low = 0.1 to M_up = 120 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_low from 0.1 to 0.08 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_low from 0.1 to 0.08 M_o?
If we change M_up from 120 to 100 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 a₃ = 2.35), determine the factor 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_up and M_low differ for the two IMFs - that is part of what makes this interesting.

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.