ASTR 400B Lab 2 -Solved

1 First Step 
Make sure to have a cloned copy of your own repository on your computer (or nimoy if you 
are using nimoy for Jupyter). Create a directory Labs/Lab2. 
From the command line git clone the class repository. If you have already done this, git 
pull to update the repository. There is a directory Labs/Lab2/ with a fifile Lab2.ipynb, which 
is the template for this exercise. 
Copy this template to your own repository directory Labs/Lab2 
2 Schechter Function 
The galaxy luminosity function in the nearby universe is well described by a Schechter 
Function: 
Φ(M)dM = (0.4 ln10) φ∗ 100.4(M∗−M)(α+1)e−100.4(M∗−M) 
dM 
(1) 
With the following parameters from Smith+2009 for Field Galaxies in SDSS at z∼0.1 in 
the Kband: 
1. φ∗ =1.66 ×10−2 h3 Mpc−3 
2. α = -0.81 
3. M∗ = M
∗
k
= -23.19 - 5log(h) 
h = the Hubble constant in units of 100 km/s/Mpc . At z=0 this is 0.7. But we are 
going to ignore it here. Units will then be in ”comoving” coordinates. 
2.1 Question 1 
Utilizing the defifined function in the template fifile, plot the Schechter Function using the 
above parameter values over a magnitude range of -17 to -26. Try to reproduce the black 
solid line in Figure 2.1, from Smith+2009 
Plotting tips: 
1. import matplotlib.pyplot as plt - this lets you use plotting functions. 
2. np.arange(0,10,0.1) will return an array from 0 to 10 spaced in intervals of 0.1 
3. plt.semilogy lets you plot the y axis as log. 
1Figure 1: Luminosity Function from Smith+2009, UKIDSS + SDSS KBand 
2.2 Question 2 
Galaxies in the Virgo Cluster have difffferent parameters, like α=-1.35 (Ferrarese+2016 ApJ 
824) Overplot the Schechter Function with this new value of α. Try a smaller value of 
α = −0.6. How does the function change? What does this mean? 
2.3 Question 3 
Build a function to compute the Schechter Function in terms of luminosity. 
Use ‘quad‘ to determine the fraction of the luminosity that lies above L* in the following 
three cases: α=-0.7 (default), α=-0.6, α=1.85. 
Schechter Function: 
Φ(L) = 
n
∗ 
L∗ 
 LL
∗ α 
e−L/L∗ 
(2) 
n∗ = 0.008 h3 Mpc−3 
L? = 1.4 × 1010L  
3 The IMF 
Create a function called Salpeter that defifines the Salpeter IMF: 
ξ(M) = ξ0(M/M  )−α 
(3) 
α = 2.35 The function should take as input an array of stellar masses, M. You will need to 
determine the normalization, ξ0, by integrating this equation over mass from 0.1 to 120 M  
2and setting the value to 1. The function should then return ξ(M), which will now represent 
the fractional number of stars. 
• from scipy.integrate import quad 
• quad(lambda x: fxn(x),xmin,xmax) 
• quad returns an array with 2 values. you want the fifirst value. 
3.1 Question 1 
Integrate your normalized function to compute the fraction of stars with stellar masses 
greater than the sun and less than 120 M  . ** Double Check: if you integrate your function 
from 0.1 to 120 you should return 1.0 
3.2 Question 2 
How might you modify the above to return the fraction of mass in stars more massive than 
the Sun? 
4 Last Step 
Git push your Lab1.ipynb fifile to your repo. Recall steps: 
1. git add fifilename 
2. git commit -m ”COMMENTS” 
3. git push 
3