Use the

*Most-reliable-points*data set that you made in Assignment 1 for this question.

a) Plot a histogram of the number of stars as a function of absolute G magnitude (

*Figure 1*)

b) Look up the G passband for Gaia (see Jordi, Gebran, Carrasco et al. 2010, A&A, 523, 48). What is the maximum and minimum stellar effective temperature, T_eff, that correspond to the upper and lower limits of this passband?

c) Since the Gaia DR1 does not provide two passbands and therefore does not provide any colours, one can do a rough estimate of how G and V bands are related by using the transformations between G and V on the Gaia web page and adopting a 'typical' main sequence value for V-I_c. Be clear about what you've adopted and where you've found the information, and give your resulting relation between G and V magnitudes.

d) Find an H-R diagram that plots V magnitude against T_eff and draw a box around the region that Gaia is sensitive to (

*Figure 2*).

e) Think about what factors may be important in forming Figure 1 and briefly discuss this curve.

PART 2 (The Effective Volume)

This question is based on prob. 3.6 in Binney and Merrifield.

a) The

*effective volume*of a survey is the ratio of the number of objects reached by the survey to the local space-density of the objects (i.e. V_eff = N/nu_0. where nu_0 is the local number density and N is the total number of stars reached by the survey). Consider the case in which the space density of stars is equal to nu_0*exp(-z/h), where |z| is the perpendicular distance to the Galactic plane and h is the scale-height of the density distribution. Take nu_0 to be constant in the plane and assume that the Sun is at the center of the Galactic plane. Show that the effective volume of a magnitude-limited all-sky survey is

V_eff = 4 pi h^3 [1/2 x^2 + (x+1)exp(-x) - 1], where x(h/pc) = 10^[0.2(m_lim - M)+1] (here ^ means exponentiation; use spherical coordinates)

b) Plot the effective volume as a function of absolute magnitude, M, for 3 different scale heights, h=100, h=200, and h=300 pc. Use the limiting magnitude of the

*Most-relable-points*data set (

*Figure 3*) and briefly discuss the result.

PART 3 (The Vertical Scale Height of the Galaxy at the location of the Sun)

a) Using the

*Most-reliable-points*data set again, we want to plot the number of stars with height, z, above or below the plane. Since the distance to the star and the Galactic latitude, b, are known for each star, you can compute the distance of the star from the Sun as projected

*along*the plane, and can also compute the height, z above or below the plane. Consider all stars within a 1 kpc-radius cylinder around the Sun and plot a histogram of the number of stars as a function of height, |z|. On the same plot, overlay a mathematical function which is a best 'by-eye' fit of this vertical distribution (

*Figure 4*). Your fit should be in the form of an exponential or sums of exponentials. Clearly indicate the functions that you've found.

b) Repeat part (a) but for all data within the 1 kpc-radius cylinder, not just the most reliable data points.

c) Compare your resulting scale heights to those found by Juric et al. 2008, ApJ, 673, 864, for SDSS data and comment on the results.