[SOLVED] ASTR400B Lab 2

Description

 

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