DUE THURSDAY MARCH 2 IN CLASS.

PART 1 (The Initial Mass Function)

The IMF (the number of stars with masses between M and M+dM) is given by

The IMF (the number of stars with masses between M and M+dM) is given by

ξ(M) ∝ M

^{-α}where M is in Solar mass units.

a) Suppose there are 100 Solar masses in total for a group of stars, i.e. N_o = 100 (see your notes for formalism) with a minimum mass of 0.08 M_sun and maximum mass of 120 M_sun. Assume a Salpeter IMF with a constant of proportionality of 0.1567 and find the total number of stars, N_tot. What fraction of the total number of stars have masses between 0.08 and 0.5 M_sun?

b) For a Kroupa (2001) IMF, what fraction of stars are in the same 0.08 to 0.5 M_sun range?

c) If you were trying to calculate the number of stars that could contribute to the ionization of surrounding gas, would the choice of a Salpeter or Kroupa IMF make a significant difference? Explain. Assume that HII regions can only be formed from O and B stars.

a) Suppose there are 100 Solar masses in total for a group of stars, i.e. N_o = 100 (see your notes for formalism) with a minimum mass of 0.08 M_sun and maximum mass of 120 M_sun. Assume a Salpeter IMF with a constant of proportionality of 0.1567 and find the total number of stars, N_tot. What fraction of the total number of stars have masses between 0.08 and 0.5 M_sun?

b) For a Kroupa (2001) IMF, what fraction of stars are in the same 0.08 to 0.5 M_sun range?

c) If you were trying to calculate the number of stars that could contribute to the ionization of surrounding gas, would the choice of a Salpeter or Kroupa IMF make a significant difference? Explain. Assume that HII regions can only be formed from O and B stars.

PART 2 (Oort's Constants)

a)

a)

*This question is based on Prob. 10.8 in Binney & Merrifield.*The vorticity of a fluid flow is defined by the following vector equation, where v is the circular velocity at the location of the Sun.
ω = ∇ x v

Use cylindrical coordinates and show that

|ω| = 2|B|

b) From the Oort's constants, A and B, given by Bovey (2017), plot the circular velocity, Vc (km/s), as a function of R (kpc), for R between 6 and 10 kpc.

c) Suppose there were a string of HII regions in a radial line between 6 and 10 kpc at time t=0. Plot the appearance of the resulting HII regions after 1E8 yrs. To do this, you may plot theta (degrees) as a function of radius, R (kpc), or plot the curved line segment in polar coordinates, whatever you choose but be clear.

d) After an HII region at R=6 kpc has travelled one complete orbit about the Galactic Center, what is theta for an HII region at 10 kpc? Look up the

d) After an HII region at R=6 kpc has travelled one complete orbit about the Galactic Center, what is theta for an HII region at 10 kpc? Look up the

*winding problem*for spiral arms and comment.PART 3 (Radial Velocity of the Circularly Rotating Galaxy from the Position of the Sun)

a) In my most recent hand-out, Equation 21 gave the radial velocity, v_r, of any object in the disk of the Milky Way as a function of its distance from the Galactic Center, R, and longitude, l, assuming circular rotation. Note that this equation was not yet expanded and so is applicable to large distances -- not just the solar neighbourhood. Plot on the same graph, v_r as a function of distance, d, from the sun for the following longitudes: 30,60,90,120,150, and 180 degrees.

b) Explain the behaviour of these curves, i.e. location of maxima, minima (if present), and positive or negative terms, where they occur etc.

a) In my most recent hand-out, Equation 21 gave the radial velocity, v_r, of any object in the disk of the Milky Way as a function of its distance from the Galactic Center, R, and longitude, l, assuming circular rotation. Note that this equation was not yet expanded and so is applicable to large distances -- not just the solar neighbourhood. Plot on the same graph, v_r as a function of distance, d, from the sun for the following longitudes: 30,60,90,120,150, and 180 degrees.

b) Explain the behaviour of these curves, i.e. location of maxima, minima (if present), and positive or negative terms, where they occur etc.